The authors consider the following nonlinear integro-partial differential equation which describes the dynamics of a single population of spiking neurons: \[ \frac{\partial u(x,t)}{\partial t} = -\sigma u(x,t) + \int^\infty_{-\infty}\omega (x-y)f(u(y,t))dy+h,\tag{1} \] where \(u (x,t)\) is a function that encodes the activity level or the average voltage of a neuronal subgroup at a position \(x\in \mathbb{R},\) at time \(t\geq 0.\) \(\omega (x)\) is a connection function that determines the coupling between subgroups, and it is assumed to be off-center coupling type. \(f(u)\) is a non-negative and non-decreasing function that denotes the neuronal firing rate or the average rate at which spikes are generated corresponding to an activity level \(u,\) and it is assumed to be a Heaviside step function. \(h\) is a constant that encodes external stimulus applied uniformly to the entire neural field. \(\sigma\) is a positive constant which denotes a rate and is assumed to be equal one.
The authors study problem (1) for spatially homogeneous coupling, and, under suitable conditions, prove the existence of a unique bump. They also study its linear stability. Furthermore, they study the spatially inhomogeneous coupling which corresponds to replacing \(\omega (x-y)\) by \(\omega (x-y) p(y)\) in problem (1); and they present analytical as well as numerical results.